Báo cáo toán học: Polar Coordinates on H-type Groups and Applications

Trong bài báo này chúng ta xây dựng các tọa độ cực vào các nhóm H-. Khi các ứng dụng, chúng tôi tính toán một cách rõ ràng khối lượng của quả bóng trong ý nghĩa của khoảng cách và liên tục trong các giải pháp cơ bản của p-sub-Laplacian vào nhóm H-loại. Ngoài ra, chúng tôi chứng minh một số kết quả không tồn tại của các giải pháp yếu cho một thoái hóa.
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Báo cáo toán học: "Polar Coordinates on H-type Groups and Applications" Vietnam Journal of Mathematics 34:3 (2006) 307–316 9LHWQD P-RXUQDO RI 0$7+(0$7, &6 ‹ 9$ 67 Polar Coordinates on H-type Groups and Applications* Junqiang Han and Pengcheng Niu Department of Applied Math., Northwestern Polytechnical University Xi’an, Shaanxi, 710072, China Received August 11, 2005 Revised November 14, 2005Abstract. In this paper we construct polar coordinates on H-type groups. As ap-plications, we explicitly compute the volume of the ball in the sense of the distanceand the constant in the fundamental solution of p-sub-Laplacian on the H-type group.Also, we prove some nonexistence results of weak solutions for a degenerate ellipticinequality on the H-type group.2000 Mathematics Subject Classification: 35R45, 35J70.Keywords: H-type group, polar coordinate, nonexistence, degenerate elliptic inequality.1. IntroductionThe polar coordinates for the Heisenberg group H1 and Hn were defined byGreiner [8] and D’Ambrosio [3], respectively. Using their introduction as in [3]we can explicitly compute the volume of the Heisenberg ball (see [6]) and theconstant in the fundamental solution of Hn (see [4, 5]). In this paper we willconstruct polar coordinates on H-type groups. In [1], the polar coordinates weregiven in Carnot groups and groups of H-type, but the expression here is slightlydifferent. As an application, we will explicitly calculate the volume of the ballin the sense of the distance and the constant in the fundamental solution of∗ The project was supported by National Natural Science Foundation of China, Grant No.10371099.308 Junqiang Han and Pengcheng Niup-sub-Laplacian on the H-type groups. Nonexistence results of weak solutions for some degenerate and singular el-liptic, parabolic and hyperbolic inequalities on the Euclidean space Rn have beenlargely considered, see [13, 14] and their references. The singular sub-Laplaceinequality and related evolution inequalities on the Heisenberg group Hn werestudied in [3, 6]. In this paper we will discuss the nonexistence of weak solutionsfor some degenerate elliptic inequality on the H-type groups. We recall some known facts about the H-type group. H-type groups form an interesting class of Carnot groups of step two inconnection with hypoellipticity questions. Such groups, which were introducedby Kaplan [9] in 1980, constitute a direct generalization of Heisenberg groupsand are more complicated. There has been subsequently a considerable amountof work in the study of such groups. Let G be a Carnot group of step two whose Lie algebra g = V1 ⊕ V2 . Supposethat a scalar product < ·, · > is given on g for which V1 , V2 are orthogonal. Withm = dimV1 , k = dimV2 , let X = {X1 , . . . , Xm } and Y = {Y1, . . . , Yk } be a basisof V1 and V2 , respectively. Assume that ξ1 and ξ2 are the projections of ξ ∈ gin V1 and V2 , respectively. The coordinate of ξ1 in the basis {X1 , . . . , Xm } isdenoted by x = (x1, . . . , xm ) ∈ Rm ; the coordinate of ξ2 in the basis {Y1 , . . . , Yk }is denoted by y = (y1 , . . . , yk ) ∈ Rk . Define a linear map J : V2 → End(V1 ): < J (ξ2 )ξ1 , ξ1 >=< ξ2 , [ξ1, ξ1 ] >, ξ1 , ξ1 ∈ V1 , ξ2 ∈ V2 .A Carnot group of step two, G, is said of H-type if for every ξ2 ∈ V2 , with|ξ2| = 1, the map J (ξ2 ) : V1 → V1 is orthogonal (see [9]). As stated in [7], it has k ∂ 1 ∂ Xj = + [ξ, Xj ], Yi , j = 1, . . . , m. (1) ∂xj 2 ∂yi i=1 ...